MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovmpt2x Structured version   Visualization version   GIF version

Theorem ovmpt2x 6774
Description: The value of an operation class abstraction. Variant of ovmpt2ga 6775 which does not require 𝐷 and 𝑥 to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
Hypotheses
Ref Expression
ovmpt2x.1 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
ovmpt2x.2 (𝑥 = 𝐴𝐷 = 𝐿)
ovmpt2x.3 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
Assertion
Ref Expression
ovmpt2x ((𝐴𝐶𝐵𝐿𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐿,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem ovmpt2x
StepHypRef Expression
1 elex 3207 . 2 (𝑆𝐻𝑆 ∈ V)
2 ovmpt2x.3 . . . 4 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
32a1i 11 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))
4 ovmpt2x.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)
54adantl 482 . . 3 (((𝐴𝐶𝐵𝐿𝑆 ∈ V) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
6 ovmpt2x.2 . . . 4 (𝑥 = 𝐴𝐷 = 𝐿)
76adantl 482 . . 3 (((𝐴𝐶𝐵𝐿𝑆 ∈ V) ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
8 simp1 1059 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐴𝐶)
9 simp2 1060 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝐵𝐿)
10 simp3 1061 . . 3 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → 𝑆 ∈ V)
113, 5, 7, 8, 9, 10ovmpt2dx 6772 . 2 ((𝐴𝐶𝐵𝐿𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
121, 11syl3an3 1359 1 ((𝐴𝐶𝐵𝐿𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  Vcvv 3195  (class class class)co 6635  cmpt2 6637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640
This theorem is referenced by:  evls1fval  19665  ptbasfi  21365  tglngval  25427  scutval  31885  igenval  33831  lcoop  41965
  Copyright terms: Public domain W3C validator